Many commercially significant applications exist for methods by which the physical properties of a microporous solid may be predicted from measurements of the pore geometry of a sample of the solid. In petroleum geophysics, for example, the maximum hydrocarbon content of reservoir rock may be estimated from measurements of the rock porosity. The fraction of the pore volume of reservoir rock occupied by water (the "water saturation") may be estimated from measurements of the rock porosity and electrical conductivity. Any method or apparatus for improving the accuracy or speed of porosity and saturation measurements would thus enhance the efficiency and reliability of reservoir description and evaluation. Any method for accurately predicting rock permeability from measurements of pore geometry would also enhance the reliability of reservoir description and evaluation. It is an object of the present invention to provide such improved methods and means for enhancing the reliability of reservoir description and evaluation.
The present invention may also be applied to predict a variety of physical properties of many types of microporous solids other than reservoir rocks. For example, the invention may be applied to predict the porosity, permeability, conductivity, and diffusivity of fluids in heterogeneous catalyst materials. For another example, the invention may be applied to predict the ionic conductivity of porous electrochemical electrodes.
Established techniques in the fields of well logging and core analysis use empirical correlations between pore volume (or pore geometry) and physical properties, such as ionic conductivity, of reservoir rock. These empirical correlations are based on the premise that the rock properties associated with pore fluids are ultimately related to the pore geometry. With this underlying premise, there have been numerous attempts to characterize pore and grain geometry in such a way that pore-dependent physical properties can be predicted from measurements of pore and grain geometry.
Most such pore and grain characterization methods are characterized by a single underlying principle: the pore space is modeled as a bundle of discrete, smooth-walled tubes with fixed or varying tortuosity, connectivity, and cross section. The smooth-walled tubes of the model may then be reduced to network models or treated as discrete elements in a mosaic partitioning of the pore space. Alternatively, the rock grains or pore space are modeled as a connected framework of line segments.
The physical measurements of pore or grain geometry carried out in performing this type of method typically involve determination of pore size distribution by mercury intrusion porosimetry, measurement of the pore aspect ratio, measurement of the internal surface area, extraction of the ellipticity of pore cross sections, counting the number and area of pores on grids of thin sections of rock, measurements of porosity by fluid displacement or Boyle's Law techniques, or characterization of rock grains by size (either volume or linear dimension, such as by sieving), ellipticity, and various shape factors.
Variations on the conventional grid technique of counting pore space area on thin sections of rock are suggested in F. W. Preston, et al., 6th Ann. Computers and Operation Research in Mineral Industries Symposium Reprint (1966), p. 1; F. W. Preston, et al., Random Processes in Geology, D. F. Merriam (Ed.) (Springer-Verlag New York/Germany 1976), p. 63; and C. Lin, "Microgeometry I: Autocorrelation and Rock Microstructure," International Assn. of Math. Geol. Journal, 14, pp. 343-370 (1982).
The Preston et al. papers, which develop a statistical description of a rock and its pore space, disclose digitizing a line across an image of a thin section of rock by assigning a -1 value for solid and a +1 value for pore space. The power spectrum of the digitized data is computed. At each point, a judgment is made as to whether a point is pore or solid so that a sharp boundary between the two is defined. The Preston, et al. method is very different from the present invention. The Preston, et al. papers do not suggest any method capable of detecting the structure observed in performing the present invention.
The Lin paper (referenced above) also suggests a method for characterizing pore space geometry by analyzing binary images of thin sections of rock. The Lin method is similar to the Preston, et al. method, with utilization of two-dimensional image statistics rather than one-dimensional statistics and computation of the auto-correlation function (closely related to the power spectrum) of the digitized data rather than the power spectrum of the data. Lin discloses generating the binary image by identifying areas of a scanning electron microscope image of a thin section as belonging either to pore space or grain space, in order to measure the correlations between a central rock grain (or pore space) and grains (or pores) at a distance from the central grain (or pore). Lin concluded that this method gave no information about anisotropy, connectedness, or correlations that significantly differed among rocks with very different transport properties. The Lin method, intrinsically different from that of the present invention, is unsuitable for measuring the structure (typically occuring at length scales smaller than on the order of 100 microns) of interest in the present invention. Indeed, the auto-correlation function, the Fourier Transform and the power spectrum of a thin section contain information about correlations between pore spaces that obscures the structural information of interest in practicing the present invention.
The use of various statistical methods to characterize individual rock grains is disclosed by J. D. Orford and W. B. Whalley, Sedimentology, 30, pp. 655-668 (1983). The Orford paper discloses using the fractal dimension (discussed below) to characterize the shapes of individual rock grains. However, the methods disclosed in the Orford paper are unsuitable for measuring the structure of an ensemble of rock grains as assembled in a rock structure. The Orford paper neither discloses nor suggests any method for measuring pore-dependent properties of a microporous solid.
The present invention differs from previously known pore and grain characterization methods in two basic respects. First, the present invention places central importance on the roughness or complexity of the pore surface, in a manner to be described in detail below. The geometric features measured according to the present invention typically have a range of sizes down to on the order of 10.sup.-2 microns or less. In contrast, the pore and grain sizes observed in the conventional measurements are typically larger than on the order of 10 microns. The conventional approximations treating pores and grains as smooth tubes or ellipses remove much of the structure observed in performing the present invention.
The second respect in which the present invention differs from conventional pore and grain characterization methods is that the present invention treats the statistics of the porous solid and its pore space as a whole. No effort is made, in practicing the present invention, to discretize the porous solid into component grains, or to divide the pore space into pores of defined average dimension. Rather, the present invention involves determining the statistical geometrical properties of the pore walls or the statistical properties of the pore space at length scales comparable to or smaller than the largest geometrical structure on the surface of the pore space.
Outside of the field of rock physics, mathematicians and physicists have developed new statistical descriptions of random media. B. B. Mandelbrot has coined the word "fractal" to describe many statistically random geometries found in nature. See The Fractal Geometry of Nature (Freeman, San Francisco, 1982) by B. B. Mandelbrot. An object possessing geometric features having a size distribution such that the number of features of size l per unit length is proportional to the size l raised to some power, p, is said to have a "self-similar" distribution of features and to be characterized by a "fractal geometry". The coast lines of islands and the perimeters of clouds are examples of fractal geometries in nature. The central goal of these researchers has been to precisely characterize geometrical properties such as the length of a coast line or the volume of a mountain range.
A variety of objects have been analyzed in terms of self-similarity or in terms of fractals. For example, the size distribution of clusters in thin metal films [see R. F. Voss, et al., "Fractal (Scaling) Clusters in Thin Gold Films Near the Percolation Threshold", Phy. Rev. Lett., 49, pp. 1441-1444 (1982); and A. Kapitulnik et al., "Percolation Characteristics in Discontinuous Thin Films of Pb", Phy. Rev. Lett., 49, pp. 1444-1448 (1982)] and the size distribution of clay particles in a clay sample [see J. Serra, Image Analysis and Mathematical Morphology, pp. 152-158 (Academic Press, New York 1982)] have been found to exhibit self-similarity. Self-similar behavior has also been observed in measuring the roughness of objects such as a fracture surface [see J. L. Chermont, et al., "Review Quantitative Fractography", J. Mater Science, 14, pp. 509-534 (1979)] and a single fine particle [see B. H. Kaye, "Specification of the Ruggedness and/or Texture of a Fine Particle Profile by its Fractal Dimension" , Powder Technology, 21, pp. 1-16 (1978) and Orford and Whalley (referenced above)].
The procedures previously used in analyzing objects in terms of self-similarity are very different from the method of the present invention. One type of such procedure (employed, for example, by Voss et al. in the above-cited reference) involves measuring the perimeter versus the area of each of a plurality of distinct objects comprising a system. An inherent difficulty with this type of procedure is delineating the boundary of each object measured. Another type of procedure measures the length of a boundary of an object whose roughness is to be characterized, as a function of various selected finite step sizes. This may be accomplished by repeatedly performing a walk about the rough object with successively smaller step size, or by covering the boundary of the rough object with objects, such as circles, of decreasing size. In the latter procedure, known as "dilation", the length of the boundary is measured to be the area covered by the circles divided by the diameter of a single circle.
Another type of procedure (suggested by Serra in the above cited reference) involves scanning the surface of a clay sample, using an electron microscope scanner, to obtain a photograph of the sample at each of several magnifications. The intensity of each photograph along a line randomly drawn thereon is measured, and the covariance of each such intensity signal is computed. From each computed covariance function, a value representing the apparent number of clay particles intersected by the line drawn on the photograph, is calculated. Serra suggests a method for determining whether or not the size distribution of the clay particles exhibits self-similarity but does not suggest how pore-dependent physical properties of interest, such as porosity and electrical conductivity, may be determined. The Serra method relies on the assumption that the ratio of the apparent number of particles intersected by each line to the actual number so intersected does not substantially change as the magnification is changed and that the intensity of the signal does not change with feature size or with magnification. The Serra method requires manipulations, including the computation of covariance functions, which are avoided in practicing the method of the present invention. The present invention involves making microscopic measurements of a microporous solid to generate feature size distribution signals very different from any signal generated by the Serra method. Furthermore, the assumption that the signal intensity does not vary with feature size or magnification made in the Serra method is inconsistent with the intensity variations observed in practicing the present invention. Application of the present invention is not limited to specific cases in which the feature size is proportional to the intensity of the signal from the feature.